A Survey and Experimental Comparison of Service-Level-Approximation Methods for NonstationaryM(t)/M/s(t)Queueing Systems with Exhaustive Discipline

Abstract

W e compare the performance of seven methods in computing or approximating service levels for nonstationary M t /M/s t queueing systems: an exact method (a Runge-Kutta ordinary-differential-equation solver), the randomization method, a closure (or surrogate-distribution) approximation, a direct infinite-server approximation, a modified-offered-load infinite-server approximation, an effective-arrival-rate approximation, and a lagged stationary approximation. We assume an exhaustive service discipline, where service in progress when a server is scheduled to leave is completed before the server leaves. We used all of the methods to solve the same set of 640 test problems. The randomization method was almost as accurate as the exact method and used about half the computational time. The closure approximation was less accurate, and usually slower, than the randomization method. The two infinite-server-based approximations, the effective-arrival-rate approximation, and the lagged stationary approximation were less accurate but had computation times that were far shorter and less problem-dependent than the other three methods.

Approximate analysis of non-stationary loss queues and networks of loss queues with general service time distributions

“…For systems with phase-type service times, we used the 'ode45' function of the MAT-LAB ODE suit to generate exact results (Shampine and Reichelt, 1997). It numerically solves the Chapman-Kolmogorov differential equations describing the system dynamics using a Runge-Kutta method, and is widely used as a benchmark (Ingolfsson et al, 2007). For the Log-Normal distribution, simulation is the only available benchmark.…”

“…For example, Davis et al (1995) showed that the performance of time-dependent loss queues is substantially influenced by the second moment and to some extent by the third moment of the service time distribution. If service time follows a phase-type distribution, one can always use a numerical ordinary differential equation (ODE) solver, like a Runge-Kutta or Euler method (Davis et al, 1995), or a faster approach like randomization (Ingolfsson et al, 2007) to compute Pr{Q(t) = i} over time. Since phase-type distributions are dense in the class of all distributions defined on non-negative real numbers (Asmussen, 2003, Theorem 4.2), one can in theory match empirical service time data sufficiently closely using a sufficiently large number of phases.…”

Performance analysis of an inbound call center with time varying arrivals

“…in Brown et al (2005), Green, Kolesar and Whitt (2007), Czachórski et al (2009) and Czachórski et al (2014) or, for the direct comparison of some examples of such methods with the numerical methods and stationary approximations, in Ingolfsson et al (2007).…”

Time-Dependent Performance Approximation of Truck Handling Operations at an Air Cargo Terminal

“…The numerical solution of the respective set of ordinary differential equations (e.g., [16,21]) and the randomization approach [10] are applicable to Markovian systems. Although these methods provide (nearly) exact results, the numerical solution is rather time-consuming [13]. Deterministic fluid approaches approximate discrete events through continuous processes.…”

“…Another class of approximations is based upon the application of steady-state models. Comparing various approximation methods, Ingolfsson et al [13] show that the stationary independent period-by-period (SIPP) approximation achieves good results within a reasonable time. This method divides the observed time horizon into multiple smaller periods and then analyzes each period independently using a stationary model [9].…”